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Expanding large global solutions of the equations of compressible fluid mechanics. (English) Zbl 1426.65152

The dynamics of moving gases in three dimensions as described by compressible isentropic Euler system is considered. Without any symmetry assumptions on the initial data, global-in-time unique solutions are constructed to the vacuum free boundary when the adiabatic exponent \(\gamma\) lies in the interval \((1, 5/3]\). The paper is organized as follows. Section 1 is an introduction. In Section 2, the stability problem is formulated and the main result in Lagrangian coordinates is stated. Vorticity bounds are explained in Section 3, main energy estimates in Section 4, and the proof of the main theorem in Section 5. The paper is finished with four appendices. In Appendix A, the properties of the background solution are summarized. Appendix B contains the basic properties of commutators between various differential operators used in the paper, while Appendix C contains the statements of frequently used Hardy-Sobolev embeddings. In Appendix D, an alternative and equivalent formulation of the problem starting from the Lagrangian coordinates are given.

MSC:

65M99 Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems
76M60 Symmetry analysis, Lie group and Lie algebra methods applied to problems in fluid mechanics
35L45 Initial value problems for first-order hyperbolic systems
35Q31 Euler equations
76N15 Gas dynamics (general theory)
76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
35R35 Free boundary problems for PDEs
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